Vectors, Scalars, and How Science Works

DOI: 10.2138/gselements.17.3.219

The Oxford English Dictionary defines ‘vector’ as a quantity having direction as well as magnitude, and ‘scalar’ as a quantity having only magnitude, not direction. Much geological research starts with fieldwork, manifestly a vector activity. In Figure 1A, the geologists are exploring the intersection of a complex, 3-D body, the layered Klokken syenite, a 4 × 3 km igneous intrusion in the Gardar alkaline province of SW Greenland, with a mountainous 3-D land-surface. I described the unusual layering in Elements v10n1 (Parsons 2014). The igneous rocks were emplaced 1,166.3 ± 1.2 million years ago, and the 650 m of 3-D topography, which reveals the inner workings of the magma chamber, was carved by the advance and retreat of the mighty Greenland ice sheet in the last few thousand years. Only the age (a U–Pb age from baddelyite, ZrO2) is a scalar quantity.

Figure 1. (A) The summit ridge of the Klokken layered syenite intrusion. Scandinavian post-docs for scale. The dark brown layers are ‘granular’ syenites and the pale layers are ‘laminated’ syenites. (B) Transmission electron microscope image of a sub-optical coherent intergrowth in a feldspar from a granular syenite collected near the bottom of (A). Structural elements are arranged to minimize coherency strain energy. The stripy phase is Na-feldspar (albite). The stripes are twins on the Albite Law. The zig-zag phase is K-feldspar (microcline); each zig and each zag is a twin. This fully coherent microstructure has retained 94% of its radiogenic 40Ar since 1,166 Ma, giving it an apparent age of 1,096 Ma. (C) Backscattered electron image of a turbid alkali feldspar from a pale laminated syenite horizon in panel A. Dark areas are albite; light areas are microcline. The area near the centre is a braid intergrowth similar to that shown in (B). It is a relic of the microtexture prior to the deuteric coarsening that led to ‘patch’ perthite. The patches are individually a mosaic of subgrains (see D). Black dots are micropores, some containing a Mesoproterozoic fluid. This is a leaky feldspar. It has retained only 57% of its 40Ar, giving it an apparent age of 662 Ma. (D) Transmission electron microscope image of albite subgrains and micropores (irregular white areas) within an albite patch similar to the dark areas in (C). The tiny white dots are ‘burn holes’ along subgrain boundaries, produced during atom-milling.

Klokken is in the Gardar Rift, famous in petrological circles because of the absence of disturbance, physical or chemical, of its ancient, mineralogically complex alkaline rocks since their emplacement. The rift includes a 1,200 m thick sequence of unmetamorphosed red sandstones. The U–Pb age was obtained as part of a NASA-sponsored project to investigate whether isotopic decay constants change over time. Based on this high-precision age, mica from a late member of the intrusion has subsequently been used to recalibrate the 87Rb decay constant employed today in Rb–Sr dating.

We use combinations of vector and scalar observations routinely in geology, but the vector–scalar interface is often more complex than appears at first sight. It has led us down some blind alleys and, sometimes, into protracted controversies. For me, personally, one of the interfaces involved so-called ‘multiple diffusion domain (MDD) noble gas thermochronology’, a method described recently in Elements by Gautheron and Zeitler (2020). With MDD, apparent ages obtained from the radioactive decay of 40K to 40Ar over time, a technique known as 40Ar/39Ar dating, and an Ar-release spectrum obtained during laboratory heating, are used to calculate the thermal history of rock samples over geological time, mostly using alkali feldspar, (Na,K)AlSi3O8.

When the method was first applied it was found that apparent ages obtained from alkali feldspars were often considerably younger than those obtained by other isotopic methods. Argon is an inert gas, and it was suggested, correctly, that leakage over geological time might be related to intergrowths known as ‘perthite’ which form when the alkali feldspar crystals cool.

Perthite is composed of two intergrown feldspar phases, one Na-rich, the other K-rich (Figs. 1B–1D). It forms because of the large difference in ionic radii between Na1+ and K1+. The feldspar structure is based on a 3-D framework of Si–O and Al–O tetrahedra, with the alkali ions in the relatively large spaces between them. At high T, the Si–Al–O framework is flexible and can accommodate Na and K ions distributed randomly, but, as it stiffens during cooling, structural strain energy can be lowered only if they form clusters. The feldspar undergoes ‘exsolution’, forming ‘perthitic intergrowths’. The scalar world of isotopic ratios and the vector world of crystal microtextures become inextricably entwined.

In 1982, Bill Brown and I began using transmission electron microscopy (TEM) and scanning electron microscopy (SEM) to study the perthitic intergrowths in alkali feldspars in the Klokken layered series (Fig. 1A), to explore their evolution with respect to cooling rate, and to understand the role and mechanisms of fluid–feldspar reactions. Using TEM, we could image lattice nodes separated by only ~0.7 nm. From mountainside to lattice: a vector scale range of >8 × 1011.

The Klokken layered series is unique (Parsons and Becker 1987). The dark brown layers (Fig. 1A) are fine-grained syenites with a granular texture. Mineral chemistry shows that they are a roof-chill series which became detached from the roof of the magma chamber in sheets that sank like giant pizzas onto pale ‘laminated syenites’ accumulating by crystal settling below. Their brown surface was produced by recent periglacial weathering; underneath, the feldspar is dark green and glass-clear in thin section. The feldspars have very fine-scale perthitic intergrowths called ‘braid’ perthite (Fig. 1B), which coarsen systematically downward, with a near-perfect log–linear relationship between their periodicity and topographic height. They are called ‘coherent’ intergrowths because lattice-scale TEM images show that the Si–Al–O framework remains continuous at the interfaces.

The pale ‘cumulate’ layers (Fig. 1A) are produced by crystal settling. They are coarse-grained, have a pronounced lamination, do not vary systematically in crystal size or mineral chemistry, and are mainly composed of white or grey turbid feldspars. The feldspars are mainly irregular ‘patch-perthites’ (Fig. 1C), formed by a dissolution–reprecipitation process called ‘deuteric coarsening’, driven by loss of coherency strain energy. The patches are composed of tiny ‘incoherent’ albite-rich or microcline-rich subgrains (Fig. 1D).

Turbidity is the norm in alkali feldspars in common plutonic rocks, such as granites. It is caused mainly by myriads of micrometre-scale, often fluid-filled, micropores and by subgrain boundaries (Figs. 1C and 1D). In the Klokken laminated layers, hydrated mafic phases, such as amphibole and mica, appear, and the upward extensions at the top of some layers are pegmatitic. The laminated syenite layers clearly acted as high-T aquifers during the cooling of the pluton, while the fine-grained, granular syenite layers with glass-clear feldspars remained impermeable.

I stumbled into noble gas thermochronology because Klokken seemed to be the ideal place to investigate the role of perthitic intergrowths in the loss of radiogenic 40Ar. We (Parsons et al. 1988) found that coherent perthitic crystals (Fig. 1B) had retained almost all their 40Ar, giving total degassing ages as old as 1,125 ± 16 Ma and maximum ages on their nearly horizontal Ar release spectra of 1,162 ± 16 Ma. The least retentive incoherent patch perthites (Figs. 1C and 1D) had apparent ages as low as 662 Ma, and strongly stepped, inclined release spectra.

The feldspars had shared the same thermal history so the nature of the interfaces between Na-rich and K-rich regions of the perthite was clearly the factor controlling Ar loss since the Mesoproterozoic. Subsequent laser-probe work (Burgess et al. 1992) confirmed that the loss of 40Ar from regions of patch perthite could be accounted for by sustained heating of subgrains for 1,166 My in the upper crust at <150 °C. All relevant microtextures formed within 0.1 My of the emplacement of the intrusion.

Multiple diffusion domain ‘thermochronology’ was introduced by Lovera et al. (1989) less than a year after Parsons et al. (1988) had appeared. Natural and laboratory Ar loss are modelled as a conceptual system of diffusion domains with simple shapes and a finite range of sizes and of Ar-loss pathways that release Ar instantaneously into a vacuum. The physical nature of the domains is inferred from the detail of the Ar release spectrum during step heating, during which the domains must remain unmodified. The domains are, effectively, scalar. Extraordinary claims for the MDD method have been made: for example, that the uplift history of Tibet can be calculated from a 40Ar/39Ar study of a single alkali feldspar sample (Richter et al. 1991). The contrast in approach to the vector, fieldwork, and microscopy-based paper of Parsons et al. (1988) could hardly be more striking.

The MDD picture of feldspar microtexture did not go down well with feldspar mineralogists and not with all Ar experts. Nevertheless, MDD modelling using feldspars became, and remains, widely applied. Joe Smith and Bill Brown, the leading experts on feldspars, sadly no longer with us, thought we needed to point out its deficiencies, which led to the appearance of Parsons et al. (1999). The snappy title is Joe’s. The publication of this paper was not straightforward, and carries an unsettling message. We submitted it to the journal that had published Parsons et al. (1988), but it was rejected on the advice of a single, anonymous, referee, who wrote:

‘In my view this is not science. The authors present a debating position paper on the value of the ‘multiple diffusion domain’ or MDD model for inverting Ar–Ar data to obtain cooling histories. There are 30 pages of text with one table and one schematic figure’.

There was no in-depth discussion of the shortcomings of the MDD method that had been systematically raised in the manuscript. I published my first paper in 1965 and this was my first and only outright rejection. I was UK editor of Contributions to Mineralogy and Petrology for 24 years. Every manuscript was reviewed by a minimum of two experts; more if they disagreed. I must have handled around 1,000 reviews. I saw none that rejected a paper in such a peremptory way. Most were kindly, thoughtful and helpful.

At a moment like this we need a philosopher, not a scientist. Here is what Karl Popper has to say on what science is and how it works (the italics are his):

‘Our theories, beginning with primitive myths and evolving into the theories of science, are indeed man-made, as Kant said ... We do try to impose them on the world, and we can always stick to them dogmatically if we so wish, even if they are false. But although at first we have to stick with our theories—without theories we cannot even begin, for we have nothing else to go by—we can, in the course of time, adopt a more critical attitude towards them. We can try to replace them by something better if we have learned, with their help, where they let us down. Thus there may arise a scientific or critical phase of thinking, which is necessarily preceded by an uncritical phase’.

Karl Popper (From: Unended Quest: An Intellectual Autobiography, Fontana Paperbacks, 1976)

Science advances in an atmosphere of carefully reasoned criticism. It moves forward when we reject theories. The Sun does not circle Earth. Phlogiston does not exist. It takes only one black swan to falsify the statement ‘all swans are white’.

References in Historical Order

Parsons I, Becker SM (1987) Layering, compaction and post-magmatic processes in the Klokken intrusion. In: Parsons I (ed) Origins of Igneous Layering. NATO ASI Series, D. Reidel Publishing Company, Dordrecht, pp 29-92

Parsons I, Rex DC, Guise P, Halliday AN (1988) Argon-loss by alkali feldspars. Geochimica et Cosmochimica Acta 52: 1097-1112

Lovera OM, Richter FM, Harrison TM (1989) The 40Ar/39Ar thermochronometry for slowly cooled samples having a distribution of diffusion domain sizes. Journal of Geophysical Research: Solid Earth 94: 17917-17935

Richter FM, Lovera OM, Harrison TM, Copeland P (1991) Tibetan tectonics from 40Ar/39Ar analysis of a single K-feldspar sample. Earth and Planetary Science Letters 105: 266-278

Burgess R, Kelley SP, Parsons I, Walker FDL, Worden RH (1992) 40Ar–39Ar analysis of perthite microtextures and fluid inclusions in alkali feldspars from the Klokken syenite, South Greenland. Earth and Planetary Science Letters 109: 147-167

Parsons I, Brown WL, Smith JV (1999) 40Ar/39Ar thermochronology using alkali feldspars: real thermal history or mathematical mirage of microtexture? Contributions to Mineralogy and Petrology 136: 92-110

Parsons I (2014) Black Swans. Elements 10: 70, 72

Gautheron C, Zeitler PK (2020) Noble gases deliver cool dates from hot rocks. Elements 16: 303-309


Much of this article is based on understanding of feldspar microtextures built up over many years in happy collaboration with a galaxy of very skilled electron microscopists. In the order in which I worked with them: Bill Brown, Richard Worden, Kim Waldron, Martin Lee, and John Fitz Gerald. Ray Burgess and Simon Kelley were central to the very informative laser-probe work. Thank you all.

1 Comment

  1. Mark Harrison on September, 2021 at 4:50 am

    True, “It takes only one black swan to falsify the statement ‘all swans are white’”. It also takes only one recrystallized K-spar (Harrison et al., 2010, EPSL 299, 426)) to falsify the statement “All relevant microtextures formed within 0.1 My of the emplacement of the intrusion”.

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